Common Logarithms (logs)

The common logarithm is the logarithm to base 10. The use of common logs made the process of multiplication and division simpler and is widely used by engineers. The numbers used with common logarithms should have the same base of 10 and raise to some power value. It can be represented in any of the following ways.

10000 = 10⁴ or ‘log of 10000 is 4’ or ‘log₁₀10000 = 4’

        The log and anti-log tables are used to convert the values between their number form and exponent form.

Log Table

Anti-Log Table

        The log is made up of two parts namely, characteristic of the logarithm and mantissa. The figure in front of the decimal point is called the characteristic and the number behind the decimal point is the mantissa. In the example log₁₀753 = 2.8768, the number ‘2’ is the characteristic and ‘.8768’ is the mantissa.

        The mantissa is always positive, whereas the characteristic may be either positive or negative. A negative characteristic will always occurs with numbers less than 1. As the mantissa is always positive, the negative sign deliberately appears as a ‘bar’ above the characteristic in order to avoid making the entire number negative.

Example

        Anti-log/anti-logarithm is a reverse process, which is are used to obtain an ordinary number from a log number. Once you get a log answer, use only the mantissa part to consult anti-log table for an answer. The characteristic part will determine the number of decimal places to move. The following arithmetic operations will explain this clearly.

Multiplication

In order to multiply any two numbers, the log values of the numbers are found from log tables and are added. The resultant log value is viewed in the anti-log tables and the answer is derived.

Example

    Solve 69.31 x 57.43

1. Log of 69.31 (using log table) = 1.841
2. Log of 57.43 (using log table) = 1.7591
3. Add the log numbers = (1.841 + 1.7591) = 3.6001
4. Convert the mantissa part (.6001) into an ordinary number (using anti-log table) = 3.981
5. Move the decimal point 3 places to the right for the final answer = 3981

Division

In order to divide any two numbers, the log values of the numbers that are found from log tables are subtracted from one another.

Example

    Solve 2500 ÷ 60

1. Log of 2500 (using log table) = 2.3980
2. Log of 60 (using log table) = 1.7782
3. Subtract the log numbers = (2.3980 - 1.7782) = 0.6198
4. Anti-log of 0.6198 (using anti-log table) = 4167
5. Move the decimal point 2 places to the right for the final answer = 41.67